Teamwork and Individual Productivity

Flores-Szwagrzak, Karol ; Treibich, Rafael

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Published in:Management Science

DOI:10.1287/mnsc.2019.3305

Publication date:2020

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Citation for published version (APA):Flores-Szwagrzak, K., & Treibich, R. (2020). Teamwork and Individual Productivity. Management Science, 66(6),2523-2544. https://doi.org/10.1287/mnsc.2019.3305

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Teamwork and Individual Productivity

Karol Flores-Szwagrzak∗ Rafael Treibich∗∗

Abstract. We propose a new method to disentangle individual from team productivity,

CoScore. CoScore uses the varying membership and levels of success of all teams to infer,

simultaneously, an individual’s productivity and her credit for each of her teams’ successes.

Crucially, the productivities of all individuals are determined endogenously via the solution

of a fixed point problem. We show that CoScore is well defined and provide axiomatic

foundations for the inferred credit allocation. We illustrate CoScore in scientific research

and sports.

Keywords: Networks, Collaboration, Productivity indices, Ranking methods

1. Introduction

In January of 2017 Futbol Club Barcelona fired Pere Gratacos for stating that Lionel

Messi, the team’s superstar, “would not be as good” if not for his superlative teammates.

Despite leading Barcelona to several national and continental championships, Messi has

always struggled with Argentina’s national team. One might then wonder if it is really Messi

who explains Barcelona’s success or rather, as Gratacos’ statement suggests, Barcelona that

explains Messi’s success. Even when coming from well informed experts, judgements of this

type are inherently subjective. The collective nature of the game means that a player’s

productivity cannot be reduced to either her individual statistics or her winning record. It

should capture the player’s contribution to the success of her team, while accounting for the

quality of her teammates. Ideally, this would require observing the counterfactual success of

the team absent a particular player, which is impossible. How can we then systematically

Date: April 11, 2019.∗Copenhagen Business School, ksz.eco@cbs.dk

hh∗∗University of Southern Denmark, rtr@sam.sdu.dkWe thank Carlos Alos-Ferrer, Sophie Bade, Miguel Ballester, Albert-Laszlo Barabasi, Salvador Barbera,Yann Bramoulle, Manel Baucells, Christopher P. Chambers, Gabrielle Demange, Enrico Diecidue, JensGudmundsson, Lene Holbæk, Ozgur Kıbrıs, Livia Helen, Jean-Francois Laslier, Isabelle Mairey, MichaelMandler, Francois Maniquet, Alan D. Miller, Juan D. Moreno-Ternero, Herve Moulin, Klaus Ritzberger,Tomas Rodrıguez Barraquer, Alexander Schandlbauer, William Thomson, Armando Treibich, Rodrigo Velez,Oscar Volij, and Lars Peter Østerdal for helpful comments. We also thank seminar audiences in Barcelona,Bordeaux, Cologne, Copenhagen, London, Louvain-la-Neuve, Lund, Maastricht, Odense, Rochester andSeoul.

1

evaluate the productivity of a player on the basis of her team’s success, that is, on the basis

of work not done entirely by herself?

The question is obviously relevant beyond sports. Many production processes require

teamwork, yet individual contributions are not perfectly, if at all, observable; this has been

the main impetus behind the study of “moral hazard” in teams (Holmstrom, 1982; McAfee

and McMillan, 1991; Guillen et al., 2014). Even formerly individual areas of production,

such as science and innovation, are increasingly dominated by teams; the vast majority of

publications are now coauthored and the average number of coauthors is steadily increas-

ing (Wuchty et al., 2007; Jones, 2009); teams also achieve higher impact research (Wuchty

et al., 2007), are more likely to avoid failures, and achieve innovation breakthroughs (Singh

and Fleming, 2010). Not surprisingly, since the individual contributions of researchers can-

not be observed, the attribution of credit for collaborative scientific research is neither well

understood nor explicit (Gans and Murray, 2013, 2014). Researchers are nonetheless eval-

uated on the basis of their increasingly collaborative research for professional advancement,

compensation, and recognition (Sauer, 1988; Bikard et al., 2015).

In this paper, we evaluate individual productivity in a simple theoretical model where team

membership and team production are observable but individual contributions towards team

production are not. The main idea is twofold: Firstly, quantifying individual productivity

requires knowing how much the individual contributes to her teams. Secondly, determin-

ing how much she contributes itself requires knowing her individual productivity since the

more productive individuals bring more expertise, team-building skills, and visibility, and

contribute more on average. This could mean that quantifying individual productivity is

impossible in the absence of information on individual contributions; conversely, it could

mean that quantifying individual contributions is impossible in the absence of information

on individual productivity. To the contrary, we show that both problems can be thought of

as dual and solved jointly.

We propose a new method, CoScore, building on this idea, by jointly scoring or co-scoring

individual performance and credit for team productivity. Individual credit on a given project

is understood as the fraction of the team output attributed to the individual for that project.

CoScore defines an individual’s productivity score as the sum of her credits over all the

projects she has contributed to. In turn, on each project, credit is allocated proportionally

to the score of each teammate. Crucially, the scores and the credits of all individuals are

determined endogenously and simultaneously as the solution of a fixed point problem.

Formally, our model consists of a database of projects C undertaken by teams composed

of individuals drawn from N . Each project p ∈ C is described by the team of individuals

T (p) ⊆ N undertaking the project and a numerical measure of output or worth w(p) ∈ R+.

2

Importantly, we take this measure as given to focus on the assessment of individual credit

and productivity. Each individual, i ∈ N , can thus be described by her production record,

Ci ⊆ C, specifying the set of projects she participated in. CoScore defines the productivity

score of each individual by the system of equations

(1) si =∑p∈Ci

si∑j∈T (p)

sjw(p) ∀i ∈ N

where si is the score of individual i. The score of each individual thus depends on the

(endogenous) scores of her teammates. CoScore is well defined whenever each individual

in the database has at least one solo project with strictly positive worth, no matter how

small: for each i ∈ N , there is p ∈ C such that T (p) = {i} and w(p) > 0. This is a

sufficient condition for the system of equations (1) to have a unique solution.1 Existence

follows from Brouwer’s fixed point theorem; uniqueness comes from the fact that a solution

of (1) maximizes a strictly concave function over an open and convex set (see Theorem 3 in

Appendix A). CoScore can be extended to tackle applications like team sports where solo

projects are not possible or where not every individual has such a project (see Section 3).

CoScore reflects the structure of the entire team membership network and the records of

possibly all teams in the database.2 The central insight is that allocating credit on a given

project requires assessing the productivity of each member in the team. Thus, the credit

assigned to an individual depends on the credit assigned to all of her teammates on all of

their projects, which is itself determined by their teammates’ credit on all of their respective

projects, and so forth. Hence, an individual’s credit on a project depends on the whole

database, not simply on how many teammates she has. This is in sharp contrast to more

naive productivity measures such as the egalitarian score, where credit is allocated equally

among the individuals in each team,

sei =∑p∈Ci

1

#T (p)w(p) ∀i ∈ N.

Measures such as the egalitarian score can only control for collaboration anonymously, based

solely on the number of teammates in a project, not on their characteristics or any other

information in the database.

1Technical assumptions of this type are common to fixed point based methods used for internet searchengines, social networks, general equilibrium, game theory, etc.

2Here, we take the team membership network to be the weighted hyper-graph where nodes representdifferent players and hyper-edges represent projects of varying weights.

3

Axiomatic analysis. The CoScore and the egalitarian score take different stances on how

to allocate credit for teamwork, resulting in different measures of individual productivity.

There are, of course, many other conceivable ways to allocate credit. We thus study the

basic problem from a general perspective, by evaluating all the possible credit allocation rules

using the axiomatic approach. Previously, the axiomatic approach has bee used extensively

to motivate otherwise ad hoc resource allocation rules (Young, 1994; Thomson, 2001). Our

axioms formalize basic properties that a credit allocation rule should obey and describe their

joint implications, singling out specific credit allocation rules.

In Section 2, we present axiomatic characterizations of the credit allocation rules associated

to the CoScore and the egalitarian score. The characterizations highlight the fundamental

differences between these rules: The egalitarian rule is the only anonymous rule ensuring

that an indivdiual’s credit depends solely on her own record, not on any other information in

the database; hence, it is the only anonymous rule that can be computed for each individual

in isolation, accounting only for the number of individuals in each project (Theorem 1). In

contrast, the axiom satisfied by the CoScore rule but not by the egalitarian rule, “aggregate

strength,” captures the idea that the inferred credit of an individual on a project should

depend on how strong she is relative to her teammates. It requires an individual’s credit in

a given project to depend on the aggregate record or “strength” of her teammates, but not

necessarily on their number. We prove that the CoScore rule is characterized by aggregate

strength and two other axioms also satisfied by the egalitarian rule (Theorem 2).

Extensions and empirical case studies. Our axiomatic analysis tackles a stylized model

where only team memberships and outputs are observable and each individual has a solo

project of strictly positive worth. In specific applications, teamwork involves further struc-

ture and observable information: the roles of teammates, their degree of participation, dif-

ferences in the time horizon of their production records, etc. Moreover, individuals may

not have solo projects as in team sports. CoScore can be extended to account for these

additional features.

To illustrate, in Section 3, we carry out two empirical case studies in scientific research

and team sports. For research, we analyse a database consisting of all articles from 33 major

economics journals over the period 1970− 2015; for sports, we analyse a database consisting

of the yearly records of all NBA teams over the period 1946− 2011.

In both cases, individuals (researchers or players) have different observable ages, some

are at the beginning of their careers while others have reached maturity. Mechanically, the

production record of senior individuals tends to be lengthier since they have had more time

to accumulate projects. To account for age, we can define the score of each individual as her4

average credit over the span of her career, instead of her total credit. This is more reflective

of individual productive ability, allowing for meaningful comparisons across individuals of

different career lengths.

In the NBA case, we also have data on the degree of participation in a team, the amount

of time each player spent on the court. It is natural to extend CoScore to incorporate this

information. We propose a family of scores parametrized by a scalar α ∈ [0, 1], where a

fraction α of credit in each project is allocated proportionally to individual time partic-

ipation and the remaining fraction (1 − α) is allocated proportionally to her endogenous

productivity weighted by her time participation. If α takes a value of one, all credit is allo-

cated proportionally to the degree of participation and we arrive at an egalitarian-like score

correcting for differences in time participations: if all time participations are equal, credit

is shared equally. At the other extreme, if α takes a value of zero, all credit is allocated

proportionally to the endogenous productivity weighted by time participation and we obtain

a score correcting for differences in the degree of participations that is closer to CoScore.

The parameter α thus quantifies the extent to which the allocation of credit depends on the

endogenous productivity. In the scientific research case, we assume project participation to

be binary since we can only observe project membership but not time participation; here a

share α of each project is allocated equally among its members while the remaining (1− α)

share is allocated proportionally to their endogenous productivity.

To determine the significance of the choice of α, we perform a sensitivity analysis assessing

how the parameter affects the productivity scores and the corresponding rankings of individ-

uals. In both applications, we observe a sharp divide between the parametric extensions of

CoScore and the egalitarian benchmark. Thus, accounting for the production records of all

team members in the attribution of credit results in markedly different scores and rankings.

Although CoScore and its extensions cannot be computed independently for each individ-

ual, since they require looking at the whole database of projects, their implementation is

straightforward. Like Google’s PageRank algorithm (Page et al., 1998), they can be applied

to extremely large databases in a systematic way, providing a concrete alternative to measure

individual productivity.

Related work. The endogenous formulation of CoScore as a fixed point and its intensive

use of the team membership network are reminiscent of various measures of network central-

ity used to rigorously quantify socioeconomic phenomena such as reputation, importance,

popularity, and intelectual influence.3 For example, “eigenvector centrality” (Bonacich, 1972)

defines the “centrality” of an individual in a social network to be proportional to the sum

3For a survey on network centrality measures see Jackson (2008) and Newman (2010).5

of the centralities of her neighbors. Such measures have proven useful in many different

contexts, as shown in particular with the PageRank algorithm used by Google to rank the

relevance of web pages (Page et al., 1998) or the recursive methods to rank journals (Pinski

and Narin, 1976; Palacios-Huerta and Volij, 2004; Demange, 2014).

Rigorous rankings of sports teams also have an endogenous nature since a team is deemed

to be “successful” not only on the basis of its wins or scores, but perhaps most importantly, if

it can defeat other “successful” teams. Like eigenvector centrality for social networks, these

rankings rely on the Perron-Frobenius theorem and fixed point methods (Keener, 1993).

CoScore relies on similar insights but differs substantially from network centrality because

it is not defined on a network but on a weighted hypergraph, the database of projects. Fur-

thermore, while the centrality of a given player increases with the centrality of her neighbors,

the credit of a given player (according to CoScore) decreases with the productivity of her

teammates.4

The application of CoScore to scientific research also contributes to a growing literature

on the measurement of individual academic productivity and intellectual influence. The

remarkable uptake of the h-index (Hirsch, 2005), now routinely reported by Google Scholar,

attests to the demand for simple indices summarizing a researcher’s publication or citation

record. It has also motivated a number of alternatives (Egghe, 2006; Lehmann et al., 2006),

some of which have been supported axiomatically (Chambers and Miller, 2014; Perry and

Reny, 2016). CoScore differs from all these measures in that it does not simply summarize

an individual record; it summarizes all records simultaneously. CoScore thus reflects the

collaborative nature of scientific production, now dominated by coauthorship (Wuchty et al.,

2007). In contrast, indices summarizing an individual’s record alone must either ignore the

fact that papers are coauthored or account for it using only paper features such as the

number of coauthors (Marchant, 2009).5

Potential applications. CoScore and the measures derived from it can also be applied to

situations where teamwork is understood more generally. For instance, it is well known that

product bundling increases revenue compared to selling the bundle’s components separately

(Adams and Yellen, 1976; McAfee et al., 1989). A bundle can then be seen as a form of

4Note, from a more technical point of view, that CoScore is not defined by a system of linear equationsand does not rely on the Perron-Frobenius theorem.

5The biases of these approaches are well known. Assigning full credit to all coauthors, artificially inflatestheir publication record by overrating the value of coauthored papers (Price, 1981; Liebowitz, 2013). Thisfavors researchers with multiple collaborators or belonging to large teams and gives perverse incentivesfor artificial coauthorship. Dividing credit uniformly corrects this bias but dilutes the credit due to theintellectual leaders of a paper, ignoring that authors’ contributions may differ substantially (Hagen, 2008;Shen and Barabasi, 2014). The same concern applies to the various proposals to allocate credit when authorsare listed by order of importance or by other field-specific conventions (Hagen, 2008; Stallings et al., 2013).

6

teamwork between its components; its price is observable and takes the role of our measure

of worth and we can asses the share of the price due to each component by observing

variations in the prices of bundles including the component. From a managerial perspective,

this can inform component purchase decisions and the design of new bundles. Similarly,

digital platforms such as Hulu, iTunes, Netflix, and Spotify charge users different access fees

for bundles of entertainment content; here, the platform manager can assess the relative

benefits and costs of including or removing content.

Museum passes or city public transportation cards also offer buyers access to bundles of

goods. Each component contributes differently to the value of the pass or card to consumers

but this is not immediately observable. Assessing the differing contributions is nonetheless

essential to fairly compensate the providers of the various components (Ginsburgh and Zang,

2003; Bergantinos and Moreno-Ternero, 2015).

Finally, a closely related problem is that of sharing sports event broadcasting revenues

among participating teams (Bergantinos and Moreno-Ternero, 2018). Here the individuals

engaging in teamwork are the sports teams and the value they produce is the broadcasting

revenue of their joint games. Analytically, the key difficulty is that teams may contribute

differently to produce the joint revenue but this cannot be observed.

2. Axiomatic Analysis

We now study credit allocation from an axiomatic perspective. We provide axiomatic

characterizations of the credit allocation rules associated to the egalitarian score and CoScore.

The axioms we introduce describe how the allocation of credit is affected by changes in

team composition and output in the database of projects. We thus start by introducing a

model allowing us to vary these parameters and thereby relate credit allocation across the

corresponding databases.

The axiomatic analysis maintains the assumption that each individual has at least one

solo project with strictly positive worth, the same condition that ensures CoScore is well-

defined. This assumption is not satisfied in some applications such as team sports. However,

the analysis here illuminates the main properties underlying CoScore and its extensions to

general real-life applications (Section 3), contrasting these with the egalitarian benchmark.6

6Similarly, the axiomatic analysis of PageRank (Altman and Tennenholtz, 2005) concerns its simplestversion which is not always well defined; a technical condition is also imposed to ensure it is (the directedhyperlink graph must be strongly connected). The axiomatic analysis of the “invariant method” (Palacios-Huerta and Volij, 2004) closely related to PageRank is also carried out under the analogous assumption thatone can get from one journal to any other journal by a directed chain of citations; this yields the “irreduciblematrix” condition necessary for the well-definedness of the invariant method.

7

2.1. Model. A database D consists of a finite collection of projects C(D). Each project

p ∈ C(D) is described by the team of individuals T (p,D) involved in it and a measure of

output or worth w(p,D) ∈ R+. We refer to p ∈ C(D) as a collaborative project when team

T (p,D) contains more than one individual. The collection of all individuals involved in

projects in database D is denoted by N(D). The production record of individual i ∈ N(D)

is denoted by Ci(D) and consists of all projects involving her, all p such that i ∈ T (p,D).

Similarly, the production record of team S ⊆ N(D) is denoted by C(S,D) and consists of

all projects involving team S, all p such that S = T (p,D). Throughout this section we

assume that each individual has at least one solo project with strictly positive yet possibly

arbitrarily small worth.

A credit allocation for project p in database D is a profile z(p) ∈ RN(D)+ distributing the

project’s output among its team members,∑

i∈T (p,D) zi(p) = w(p,D). We thus exclude the

possibility that, for example, two individuals each receive more than half of the credit for

a project.7 This is, however, without loss of generality and our results can be modified to

reflect more general assumptions whereby the total credit on a collaborative project depends

on the team’s size and output.8 We denote by Z(p,D) all the possible credit allocations for

project p in database D . A credit allocation for database D specifies a credit allocation for

each project in the database; we denote by Z(D) the possible credit allocations for database

D , Z(D) ≡ ×p∈C(D)Z(p,D).

A database involving three individuals and a corresponding credit allocation are illustrated

in Figure 1. Projects are represented by rectangles of varying sizes proportional to their

worth. A credit allocation specifies the share of credit on each project due to each team

member; the credit of a member can thus be represented as the area within a rectangle

assigned to her. A particular credit allocation can then be illustrated as a “colouring” of

each of the rectangles as depicted in Figure 1 (ii). Note that the share of credit allocated to

each individual need not a priori be the same across projects, even if these involve the same

team; for instance, in Figure 1 (ii), B gets a larger share of credit in her larger joint project

with A.

2.2. Rules. A (credit allocation) rule recommends a credit allocation for each database.

Formally, a rule r is a function mapping each database D into a credit allocation r(D) ∈

7Instead of being part of the definition of a credit allocation, this assumption could instead be explicitlystated as an axiom and the ensuing analysis would be unchanged.

8For example, Bikard et al. (2015) argue that the implicit attribution of credit in scientific research isconsistent with valuing an n-member project at

√n times its output. In scientific research, a project is a

paper, the team undertaking it is constituted by its coauthors, and its output or worth is understood as itsnumber of citations.

8

CC A

A

AA

A

B

B

B

B

B

(i) Database: Projects are represented by rect-

angles of varying sizes proportional to their worth.

Ann has two solo projects of worth 16 and 2, Bobhas two solo projects of worths 4 and 1, Cat has one

solo project of worth 18. Ann and Bob have two

joint projects of worths 36 and 14. Finally, Ann,Bob and Cat have a joint project of worth 45.

CC A

A

AA

A

B

B

B

B

B

(ii) Credit allocation: The credit allocated to

each individual is represented by the corresponding

shaded area. The credit for each of the solo projectsgoes entirely to the corresponding individual. The

total credit allocated on a joint project equals the

project’s worth.

Figure 1. Database and credit allocation rule.

Z(D). We denote by ri(p,D) the credit assigned by rule r to individual i for project p in

database D and by r(p,D) the profile (ri(p,D))i∈N(D).

The most basic rule, the egalitarian rule, allocates the worth of each project among its

team members equally, irrespective of all other information available in the database: for

each database D , each p ∈ C(D), and each i ∈ T (p,D),

ri(p,D) =1

#T (p,D)w(p,D).

The CoScore rule allocates credit on each project proportionally to the productivity score

of each member of the team, as measured endogenously by the CoScore in the relevant

database: for each database D , each p ∈ C(D), and each i ∈ T (p,D),

(2) ri(p,D) =si(D)∑

j∈T (p,D)

sj(D)w(p,D).

where si(D) denotes the score of individual i in database D as measured by CoScore,

(3) si(D) =∑

p∈Ci(D)

si(D)∑j∈T (p,D)

sj(D)w(p,D) ∀i ∈ N(D).

The CoScore rule is well defined because CoScore is (see Theorem 3 in Appendix A).

Remark 1. The CoScore rule can also be defined endogenously: For each database D and

each individual i ∈ N(D), replace si(D) with∑

p∈Ci(D) ri(p,D) in the definition of the

CoScore rule in equation (2).9

When there are only two individuals in the database, the CoScore rule allocates credit on

joint projects proportionally to each individual’s aggregate solo contribution. When the

database contains more than two individuals, the CoScore rule does not generally admit a

simple closed form representation. A notable exception is the case where each joint project

in the database involves all individuals; there, the credit on joint projects is also allocated

proportionally to each individual’s aggregate solo contribution. Figure 2 illustrates both the

egalitarian and the CoScore rules in the two-individual case.

B

B

A

A

(i) Allocation of credit according to the egalitarian

rule. A and B receive equal credit for their jointproject.

B

B

A

A

(ii) Allocation of credit according to the CoScore

rule. The ratio between the credit received by Aand B on their joint project is the same as the ratio

between the worth of A’s solo project and B’s solo

project.

Figure 2. Egalitarian and CoScore rules for a database with 2 individuals.

2.3. Axioms and characterizations. We first introduce the two axioms characterizing

the egalitarian rule. The first axiom, “anonymity,” requires credit to be allocated on the

basis of information observable in the database only; exogenous characteristics associated

to each individual’s “name tag” should bear no influence. Thus, if we could switch the role

of individuals i and j in a database by permuting their production records, then the credit

assigned to i on each project after the permutation should be the same as the credit which

was assigned to j before it, and conversely.

Anonymity: For each pair D ,D ′, if C(D) = C(D ′), w(p,D) = w(p,D ′) for each p ∈ C(D),

and π : N(D) → N(D ′) is a bijection such that T (p,D ′) = π(T (p,D)) for each p ∈ C(D),

then ri(p,D) = rπ(i)(p,D′) for each i ∈ N(D) and each p ∈ C(D).

Anonymity is a basic property excluding arbitrary credit allocation rules. For example, it

precludes a predetermined individual from receiving all credit on each of her collaborative

projects. The egalitarian and CoScore rules are anonymous.

The second axiom, “own record only,” requires that an individual’s credit on a project

should depend only on her own production record. Her credit on a project may however

depend on the composition of her team, not just on its size. To state the axiom formally, we10

specify what it means for two collections of projects to be the same across two databases.

For each pair D,D′, each C ⊆ C(D), and each C ′ ⊆ C(D′), we say that C and C ′ coincide

if C = C ′ and, for each p ∈ C, w(p,D) = w(p,D′) and T (p,D) = T (p,D′).

Own record only: For each pair D,D′ and each i ∈ N(D) ∩ N(D′), if Ci(D) and Ci(D′)

coincide, then, for each p ∈ Ci(D), ri(p,D) = ri(p,D′).

The axiom requires that, if an individual has the same production record in two databases,

then her credit on each of her projects in both databases is the same. Thus, a rule satisfies

own record only if the allocation of credit to an individual remains unaffected by changes

in the database that do not alter her production record. Clearly, the CoScore rule does

not satisfy this axiom because the score of each individual depends possibly on all of the

information in the database.

We can now present the characterization of the egalitarian rule.

Theorem 1. A rule satisfies anonymity and own record only if and only if it is the egalitarian

rule.

The properties in Theorem 1 are logically independent. See Appendix D.

We now introduce the three axioms characterizing the CoScore rule. The first axiom

specifies that a rule’s credit allocation behavior is “consistent” in the following sense: Upon

removing an individual from the database, keeping fixed the characteristics of each project

not involving her, the allocation of credit among the remaining individuals is unaffected. Re-

moving an individual obviously requires removing her individual projects from the database.

However, removing her collaborative projects as well would artificially reduce her teammates’

production records. On the other hand, leaving her collaborative projects in the database,

absent her participation, would artificially inflate her teammates’ contribution. Thus, all of

her collaborative projects should stay in the database, but their worth should be updated

by subtracting the credit that was originally assigned to her.

To state the axiom formally, we now define the “reduced database” obtained after removing

an individual from a database given a credit allocation. For each database D , each credit

allocation x ∈ Z(D) and each individual k ∈ N(D), we denote by D−kx the database obtained

from D by removing all of k’s non-collaborative projects and, for each of k’s collaborative

projects, by withdrawing her from its team and reducing its worth by the credit assigned to

her under x. Thus, C(D−kx ) = C(D) \ C({k},D) and, for each collaborative p ∈ Ck(D),

T (p,D−kx ) = T (p,D) \ {k} and w(p,D−kx ) = w(p,D)− xk(p).11

The team and worth of each project not involving k are identical in D−kx and D .

Consistency: For each k ∈ N(D), each i ∈ N(D) \ {k}, and each p ∈ C(D−kr(D)),

ri(p,D) = ri(p,D−kr(D)).

Thus, rule r is consistent if the credit allocation for all individuals other than k is the

same in the reduced database D−kr(D) as in the original database D . Palacios-Huerta and

Volij (2004) use a similar consistency property in their axiomatic characterization of the

“invariant method” to rank academic journals. Their “reduced” journal database is obtained

by reassigning the citations from the withdrawn journal in a predetermined way. In contrast,

we readjust the worth of each project endogenously, since the reduction depends on the

credit allocated to the withdrawn individual by the rule itself.9 Consistency properties have

been used extensively in resource allocation and game theory (see Thomson, 2011, 2016, for

surveys). The egalitarian and the CoScore rules are consistent. Consistency is illustrated in

Figure 3 for a database with three individuals.

CC A

A

AA

A

B

B

B

B

B

(i) Allocation of credit in the originaldatabase.

CC A

A

AA

A

B

B

B

B

B

(ii) Reduced database after Cat is re-moved.

A

A

AA

A

B

B

B

B

B

(iii) Consistency: Ann and Bob’scredit on each of their projects remains

unaffected.

Figure 3. Consistency

The second axiom, “aggregate worth,” captures the idea that the credit allocation should

only depend on the aggregate value produced by each team, not on this value’s breakdown

into projects of different worth. It requires that, for example, replacing two projects of worth

100 and 200 by a single project of worth 300, both involving the same team, does not affect

the credit allocation.

Before stating the axiom formally, we define the database obtained after replacing a set

of projects involving a team by a single project of equal aggregate worth involving the same

team. For each database D , each team S ⊆ N(D), and each set of projects involving that

team C ⊆ C(S,D), we denote by DC the database obtained by replacing all projects in C

9See Dequiedt and Zenou (2017) for an application of a consistency property in the characterization of aclass of network centrality measures.

12

by a single project pC involving team S and with total worth equal to that of the sum of all

projects in C, all else equal.10

Aggregate worth: For each C ⊆ C(D) such that all projects in C involve the same team,

r(pC ,DC) =∑

q∈C r(q,D) and, for each p ∈ C(DC) \ {pC}, r(p,DC) = r(p,D).

Thus, a rule satisfies aggregate worth if the credit allocated to each team member for the

aggregate project pC in the new database DC is equal to the total credit that was previously

allocated to her for all projects in C, and the allocation of credit on all other projects is

the same as before. Aggregate worth is satisfied by the egalitarian and the CoScore rules.

Aggregate worth is illustrated in Figure 4 for a database with three individuals.

CC A

A

AA

A

B

B

B

B

B

(i) Allocation of credit in the originaldatabase.

CC A

A

AA

B

BB

B

(ii) Reduced database after Ann andBob’s two joint projects are aggre-

gated into a single project of equal to-

tal worth.

CC A

A

AA

B

BB

B

(iii) Aggregate Worth: Ann andBob’s credit on the aggregate project

are equal to the total credit they were

allocated on each of their two jointprojects. The allocation of credit on

all other projects remains unaffected.

Figure 4. Aggregate Worth

The last axiom, “aggregate strength,” captures the idea that an individual’s credit on

a project should depend on how strong she is relative to her teammates, not just on how

many teammates she has. More precisely, it requires the individual’s credit to be unaffected

if a group of her teammates on the project are replaced by a single “aggregate” teammate

whose record comprises all the projects that previously involved members of that group. The

individual’s credit should depend only on her own record and the aggregate record of her

teammates in that project, since their aggregate record reflects their overall strength.

The axiom only aggregates groups of teammates sharing the same collaborative projects,

as it would be questionable to require the allocation of credit to remain unaffected if the

collaborative projects involving the group varied considerably. Assume for instance that in-

dividual C has two projects of identical worth, one with individual A and one with individual

10Formally, for each S ⊆ N(D) and each C ⊆ C(S,D), database DC comprises the collection of projectsC(DC) = [C(D) \C]∪ {pC}, where project pC involves each member of S, T (pC ,DC) = S, and has a worthequal to the total worth of the projects in C, w(pC ,DC) =

∑p∈C w(p,D). For each project in C(D) \C, its

team and worth are identical in D and DC .13

B ; A has a “very weak” record while B has a “very strong” record.11 Therefore, individual

C should get a larger share of the credit on her project with A than on her project with B

since B is much stronger than A. Now, suppose we replaced A and B by an individual with

their aggregate records and required C ’s credit to be unaffected. Then C ’s credit on both

projects would be different despite these projects now being identical (involving the same

teammates and having the same worth).

To state the axiom, we define the database resulting from the replacement of a group by an

individual with the combined project record of the members of the group. For each database

D and each group of individuals S ⊆ N(D) with the same collaborative projects, we denote

by DS the database obtained by replacing all the members of S by a single individual denoted

by iS with the combined record of the members of S, all else equal.12

Aggregate strength: For each S ⊆ N(D) such that each individual in S has the same

collaborative projects and each i ∈ N(D) \ S, ri(DS) = ri(D).

Thus, a rule satisfies aggregate strength if the allocation of credit for the individuals outside

S is identical after the members of S have been replaced by individual iS. An implication is

that the credit allocated to iS for each project must be equal to the total credit that used

to be allocated to the members of S. Aggregate strength is satisfied by the CoScore rule

but not by the egalitarian rule. Aggregate strength is illustrated in Figure 5 for a database

with three individuals; observe that the aggregated individuals, A and B, have the same

collaborative projects.

CC A

A

AA

A

B

B

B

B

B

(i) Allocation of credit in the original

database.

CC D

D

DD

D

D

D

(ii) Reduced database after Ann and

Bob are “aggregated” into Dan.

CC D

D

DD

D

D

D

(iii) Aggregate Strength: Cat’s credit

on all her projects remains unaffected.

Figure 5. Aggregate Strength

11 For example, beyond their collaborative project, their records could consist solely of individual projects,with each of B ’s projects having a greater worth than each of A’s. Of course, the intuition applies moregenerally.

12Formally, DS comprises the same projects as D , C(DS) = C(D); for each project p ∈ C(DS), its worthis the same in the two databases, w(p,DS) = w(p,D) and, if no member of S belongs to T (p,D), its teamis also the same T (p,DS) = T (p,D), otherwise T (p,DS) = [T (p,D) \ S] ∪ {iS} so the members of S arereplaced by individual iS .

14

We can now present the characterization of the CoScore rule.

Theorem 2. A rule satisfies consistency, aggregate worth, and aggregate strength if and only

if it is the CoScore rule.

The properties in Theorem 2 are logically independent. See Appendix D.

Recall that the egalitarian rule also satisfies consistency and aggregate worth while the

CoScore rule satisfies anonymity; these axioms are common to both rules. The difference

between the rules thus boils down to that between own record only and aggregate strength.

3. Extensions and case studies

3.1. Extensions. CoScore can be extended to applications where individuals may not have

solo projects and to incorporate additional observable information sharpening the assessment

of individual productivity and credit. To illustrate, we introduce a family of scores extending

CoScore that we will apply in our empirical case studies. This family is not meant to be the

most general but simply to illustrate the potential enrichments of CoScore.

The additional observable information we consider are the ages of individuals and the time

they participate in each project. For expositional ease, we henceforth drop the database index

D , using the lighter notation of the Introduction. For each individual i ∈ N , we denote her

age by Ai and, for each project p ∈ C, we denote by ti(p) the amount of time she participated

in the project. We assume that ages are strictly positive and that ti(p) > 0 if and only if i

belongs to T (p). When the time individuals participate in a project cannot be observed, as

in a coauthored scientific paper, we refer to participations as binary adopting the convention

that, for each individual i ∈ N and each project p ∈ C, ti(p) = 1 if i belongs to T (p) and

ti(p) = 0 otherwise.

Each member of the family of scores defined next is parametrized by a scalar α ∈ [0, 1];

we call the score corresponding to parameter α the generalized α-CoScore. It allocates a

fixed share α of the credit on each project proportionally to the time participation of each

teammate while the remaining share (1− α) is allocated proportionally to each teammate’s

endogenous score weighted by her time participation. The score is then defined as the average

credit over the length of the individual’s career. Formally, the generalized α-CoScore defines

the average productivity score of each individual by the system of equations

(4) si =1

Ai

∑p∈Ci

α ti(p)∑j∈T (p)

tj(p)+ (1− α)

ti(p)si∑j∈T (p)

tj(p)sj

w(p) ∀i ∈ N.

15

The generalized α-CoScore is well defined whenever α is strictly positive. Alternatively, if

α equals zero, this can be ensured maintaining the assumption that each individual has at

least one individual project with strictly positive worth. These are sufficient conditions for

the system of equations (4) to have a unique solution (see Theorem 3 in Appendix A).

In fact, the well-definedness of CoScore depends critically on the presence of a positive

constant term (that may be arbitrarily small) in the right-hand side of the equalities in its

defining system of equations. For example, modifying (1) by adding any strictly positive

term εi as follows

si = εi +∑p∈Ci

si∑j∈T (p)

sjw(p) ∀i ∈ N

is sufficient to guarantee the above system has a unique solution. Similarly, the first term

inside the summation in (4) is a strictly positive constant when α > 0 and ensures well-

definedness. (See the proof of Theorem 3 in Appendix A.)

The formulation of the generalized α-CoScore is flexible, including several interesting

special cases:

(1) CoScore corresponds to the case where α = 0, all individuals have the same age

normalized to one, and project participations are binary.

(2) The egalitarian score corresponds to the case where α = 1, all individuals have the

same age normalized to one, and project participations are binary.

(3) When α = 0, the score allocates credit proportionally to the individual’s expected

contribution given her time participation ti(p)si since si is interpreted as the individ-

ual’s average productivity. We refer to this case as the generalized CoScore.

(4) When α = 1, credit is allocated proportionally to time participations. We refer to this

case as the generalized egalitarian score. We use the term “egalitarian” since credit

is allocated equally whenever all team members have the same time participations.

In practice, an adequate choice of αmay reflect the structure of teamwork in an application.

Low values of α may be appropriate where individual contributions determine team success.

Credit should then be allocated mostly on the basis of individual quality. In contrast, when

a team is only as strong as its weakest member, high α values and a correspondingly more

egalitarian allocation may be appropriate. For example, success in football depends relatively

more on an all round strong team than it does in basketball (as argued by Anderson and

Sally, 2013) and this may call for a higher α in football.16

3.2. The productivity of academic researchers. Over the past thirty years, coauthor-

ship has come to dominate the production of economic research (Laband and Tollison,

2000; Card and DellaVigna, 2013; Hamermesh, 2013), reflecting the general trend in science

(Wuchty et al., 2007). A particular characteristic of economics is that authors are over-

whelmingly listed alphabetically and that the specific role of each author on a paper is not

specified.13 Not surprisingly, economists have expressed concerns on how to properly eval-

uate increasingly collaborative researchers (see, for example, Hamermesh, 2013; Liebowitz,

2013). This motivates the application of CoScore to the field of economics.

A project is now an academic paper, its members are the coauthors of the paper and its

worth is the number of citations of the paper. For each paper, an author’s participation is

binary, with value 1 if she is an author on the paper and 0 otherwise. We use Thomson

Reuters’s Web of Science database to extract all published articles with at least one cita-

tion from 33 major journals in Economics over the period 1970 − 2015 (see Appendix E).

The database includes 73732 articles, for a total of 29226 different authors. The share of

coauthored papers increases steadily from only 20% in 1970 to almost 80% in 2015, with the

average number of coauthors going from 1.25 to 2.31. Papers with more than two authors, a

rare occurence before the nineties, represent almost 40% of all papers published in 2015. The

increasing prominence of coauthorship also translates in the fraction of coauthored citations,

which exhibits the same trend as the fraction of coauthored papers. Figure 6 shows the

evolution of the fraction of coauthored papers and citations since 1970.

1970 1985 2000 2015

0.2

0.4

0.6

0.8

Year

Fractionof

coau

thored

pap

ers

n ≥ 4n = 3n = 2

1970 1985 2000 2015

0.2

0.4

0.6

0.8

Year

Fractionof

coau

thored

citation

s

n ≥ 4n = 3n = 2

Figure 6. Fraction of coauthored papers (left) and citations (right) with 2, 3 and4 or more authors from 1970 to 2015.

We compute the generalized α-CoScore by repeatedly iterating the fixed point operator

φ : x ∈ RN++ 7→ φ(x) ∈ RN

++ defined by

13 Engers et al. (1999) show how alphabetical name ordering, as opposed to relative contribution order-ing, might emerge at equilibrium when coauthors bargain over the ordering after observing their relativecontributions. In a recent paper Ray and Robson (2018) argue in favor of what they call certified random-ization, where coauthors are either listed in order of contribution, or in random order. In the latter case,the randomization is made explicit by using a specific symbol between the name of the authors.

17

(5) φi(x) =1

Ai

∑p∈Ci

α 1

#T (p)+ (1− α)

xi∑j∈T (p)

xj

w(p) ∀i ∈ N

starting from a vector of equal scores. The process converges quickly to the desired fixed

point. We also compute the generalized egalitarian score (the generalized 1-CoScore), where

the full credit on each paper is allocated equally between coauthors.

We observe a sharp divide between the generalized α-CoScore (for α < 1) and the gener-

alized egalitarian score. Figure 7 shows the cumulative distributions for the differential in

ranks and scores between the generalized α-CoScore and the generalized egalitarian score

for α = 0.1, 0.3, and 0.5. Accounting for the strength of coauthors in a given paper, not only

their number, yields significant differences both in rankings and scores. If we focus on the

12557 authors who have at least one citation per year on average (all the authors who have

an egalitarian score of at least one), individual ranks differ on average from the egalitarian

ranks by 798.5, 1342.6 and 2345.6 positions, respectively, while individual scores differ on av-

erage by respectively 16.4%, 25.3% and 37.3%. Interestingly, we observe that only a limited

number of authors (15.4%, 15.9% and 17.1%, respectively) increase their score compared to

the egalitarian benchmark.

−8,000 −6,000 −4,000 −2,000 0 2,000 4,000 6,0000

0.2

0.4

0.6

0.8

1

∆ Rank =EgRank-CoRank

P(<

)

α = 0.1

α = 0.3

α = 0.5

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

∆ Score =EgScore-CoScore (% EgScore)

P(<

)

α = 0.1

α = 0.3

α = 0.5

Figure 7. Distribution of the rank and score differentials between the generalizedegalitarian score and the generalized α-CoScore. Differences in scores are expressedin percentage of the generalized egalitarian score.

In order to understand the determinants of the generalized α-CoScore, we run OLS re-

gressions of the individual’s total share of credit on coauthored papers (as allocated by the

generalized α-CoScore) on:

(i) the individual’s average number of coauthors on joint papers (# coauthors/paper),

(ii) the individual’s total number of different coauthors (# coauthors),

(iii) the individual’s share of solo citations, and18

(iv) the log of the ratio between the individual’s generalized egalitarian score (Egali) and

the average generalized egalitarian score of her coauthors (EgalNi).

The ratio Egali/EgalNiprovides a first approximation of the individual’s strength relative

to her coauthors.14

Table 1. Regression analysis: allocated share of credit for the generalized α-CoScore

(1) (2) (3) (4) (5)

0.1-CoScore 0.3-CoScore 0.5-CoScore 0.7-CoScore 0.9-CoScore

# coauthors/paper -0.0791∗∗∗ -0.0871∗∗∗ -0.0941∗∗∗ -0.100∗∗∗ -0.106∗∗∗

(-37.86) (-66.41) (-104.67) (-147.23) (-174.17)

# coauthors 0.00512∗∗∗ 0.00358∗∗∗ 0.00232∗∗∗ 0.00124∗∗∗ 0.000313∗∗∗

(25.35) (28.25) (26.69) (18.88) (5.31)

Solo share 0.153∗∗∗ 0.0968∗∗∗ 0.0597∗∗∗ 0.0330∗∗∗ 0.0134∗∗∗

(30.56) (30.82) (27.73) (20.23) (9.15)

log(Egali/EgalNi) 0.150∗∗∗ 0.111∗∗∗ 0.0767∗∗∗ 0.0452∗∗∗ 0.0161∗∗∗

(132.81) (156.69) (157.48) (122.29) (48.65)

Constant 0.604∗∗∗ 0.632∗∗∗ 0.656∗∗∗ 0.677∗∗∗ 0.695∗∗∗

(108.09) (180.43) (273.18) (371.42) (426.53)

Observations 12134 12134 12134 12134 12134

Adjusted R2 0.724 0.793 0.818 0.808 0.760

t statistics in parentheses

∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

The regression results indicate that lower values of α generally favor authors who have

proved their individual quality by having published either a large fraction of single authored

papers, with multiple groups of coauthors and/or with relatively weaker coauthors. However,

as the following examples illustrate, authors might sometimes benefit from lower values of

α even if some (or all!) of these conditions are not satisfied. For instance, author A,15 who

has 74 papers for a total of 8113 citations, goes from being ranked 89th with the generalized

egalitarian score to being ranked 41st with the generalized α-CoScore (for α = 0.1), while

her score is almost doubled, despite a very small fraction of single authored citations (0.1%).

14A detailed decription of these variables is provided in the appendix.15Authors A, B and C mentioned below are real authors taken from the database.

19

In this case, author A benefits from having worked with a large number of coauthors (66

coauthors), thus showing her ability to produce high quality work independently of the

strength of a given coauthor. Author B, who has written 73 papers for a total of 7035

citations together with 27 different coauthors and has more than 50% of single authored

citations, satisfies all the conditions mentioned before. Yet, her generalized α-CoScore (for

α = 0.5, 0.3 and 0.1) is inferior to her generalized egalitarian score, and her ranking drops

from 59th to respectively 82th, 90th and 97th because some of her most cited papers are

coauthored with author C, a Nobel prize winner and ranked respectively 3rd, 4th and 4th

according to the α-CoScore. Although having a large number of coauthors or a strong

individual record is usually beneficial, it does not necessarily lead to an improved score over

the generalized egalitarian score. What matters ultimately is the strength of a researcher’s

coauthors. Figure 8 illustrates the coauthorship network of author C.

3.2.1. Beyond economics. In economics or mathematics authors are listed alphabetically.

However, in many other fields, the authorship order reflects either the importance or the

nature of their contribution to the paper. In the biological sciences, for example, most

recognition typically goes to the first and last authors, while the remaining authors’ con-

tributions can be harder to determine. In other fields like marketing, the convention is not

clear (Maciejovsky et al., 2009). Increasingly, journals require the capacity of each coauthor

(research design, research, data analysis, writing, etc) to be specified (Allen et al., 2014).

Although conventions vary widely, even across sub-fields, this information is indicative of

the contribution of each author and should play a role in the allocation of credit.

Our analysis can naturally be extended to account for ordered authorship and the capacity

of each coauthor. A first alternative is allocating relatively more credit to authors who have

a higher rank or capacity, by introducing paper-specific weights in the right hand side of the

CoScore formula. A second alternative is separating the allocation of credit across groups of

authors who have contributed to the paper in similar capacities.16 Each paper is subdivided

into several artificial papers, one for each group, where the number of citations assigned

to each paper (or group) depends on the relative importance of the associated capacity.17

CoScore and its extensions can then be applied to the resulting enlarged database.

16It is common for a paper to state that several of its authors contributed in the same capacities andfor its authors not to be ranked strictly, for example, having two or more lead authors. A paper’s authorscan thus be ordered into contribution classes, whereby authors in the highest class are considered the mostsignificant contributors, authors in the second highest class are the second most significant, and so forth.

17The credit allocation across groups of authors could follow the proposals of Stallings et al. (2013) orHagen (2008).

20

Figure 8. Coauthorship network of author C. Lines represent collaborations be-tween pairs of authors, darker lines reflecting a higher number of joint citations.Nodes represent authors whose degree of separation from author C (at the center)is at most 2. For each author (i) the size of the inner gray circle is proportionalto the number of single-authored citations, (ii) the size of the outer gray circle isproportional to the total number of citations and (iii) the size of the orange ring isproportional to the number of jointly authored citations credited to the author bythe generalized α-CoScore (for α = 0.1).

3.2.2. Beyond citations as a measure of worth. Following a long tradition (Garfield, 1972),

we measure the quality of a scientific paper by its total number of citations. Citations provide

an objective metric that is based on observable data and can be used systematically on a

large scale.18 However, citations have been shown to suffer from various shortcomings: (i)

citation patterns and intensities differ substantially across fields and even sub-fields, making

inter field comparisons problematic (Radicchi et al., 2008; Ellison, 2013), (ii) citations take

time to accumulate, underestimating the value of recent papers (Wang et al., 2013), and

(iii) citation counts treat each citation equally, ignoring that citations originating from more

18Perhaps the focus on citations is due to the lack of alternatives. Expert evaluations have been shownto suffer from systematic biases (Boudreau et al., 2016).

21

influential papers reflect a higher value. In response, various alternative metrics have been

proposed, respectively: (i) normalizing the citation count by the average per-paper citations

in the field (sub-field) to insure comparability (Radicchi et al., 2008), (ii) measuring the

quality of the journal where the paper was published using, for example, the impact factor

or some other metric (Palacios-Huerta and Volij, 2014), (iii) discounting the citation count

by the age of the paper, and (iv) giving more weight to citations originating from more

influential papers, as measured by recursive network centrality measures (Pinski and Narin,

1976; Palacios-Huerta and Volij, 2004). CoScore and its extensions can be computed with

any of these alternative measures of paper quality, providing complementary assessments of

a scientist’s productivity.

3.3. The productivity of sports players. We focus on the National Basketball League

(NBA) because it consists of a closed pool of players and teams, which means it does not

require comparing outcomes (wins) across leagues of possibly uneven value.19 We construct

an exhaustive database consisting of the yearly records of all NBA teams from 1946 (the

creation of the league) until 2011. A project now corresponds to a team for a given season

(e.g. Boston Celtics in 2007/2008), its members are the players who played on the team that

given season and its worth is the number of winning games in the season.20 For each project

(team.season), a player’s participation corresponds to the number of minutes spent on the

court for the team during that season. The database consists of 1352 projects (team.season),

for a total of 3800 different players.

We compute both the generalized egalitarian score and the generalized α-CoScore for

parameters α = 0.1, 0.3, and 0.5. As in the previous application, we observe significant

differences both in terms of rankings and scores. Individual ranks (compared to the general-

ized egalitarian ranks) differ on average by respectively 72.5, 122.6 and 203.9 positions, while

individual scores differ on average by respectively 32.6%, 51.3% and 79.4%. Figure 9 shows

the (cumulative) distributions for the differential in ranks and scores between the generalized

α-CoScore (for α = 0.1, 0.3 and 0.5) and the generalized egalitarian score. As in the previous

illustration, we observe that only a limited number of players increase their scores compared

to the egalitarian benchmark (14.2%, 13.1% and 11.5%, respectively). These are the true

most valuable players (MVPs) of the league, players who bring the most value to their teams

and can lead them to victory even if they are not surrounded by other star players.

19In football, for example, players often participate in leagues of varying quality. The application ofCoScore would then first require defining a scale so as to meaningfully compare wins across leagues.

20For simplicity, we do not account for playoff wins as this would require additional assumptions regardingtheir weight relative to wins in the regular season.

22

−1,000−800 −600 −400 −200 0 200 400 600 800 1,000 1,200 1,4000

0.2

0.4

0.6

0.8

1

∆ Rank=EgRank-CoRank

P(<

)

α = 0.1

α = 0.3

α = 0.5

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

∆ Score =EgScore-CoScore (% EgScore)

P(<

)

α = 0.1

α = 0.3

α = 0.5

Figure 9. Distribution of the rank and score differentials between the generalizedegalitarian score and generalized α-CoScore. Differences in scores are expressed inpercentage of the generalized egalitarian score.

We run OLS regressions of a player’s share of credit under the generalized α-CoScore

on (i) the number of different teammates she has had (normalized by the length of her

career) and (ii) the ratio between her generalized egalitarian score (Egali) and the average

generalized egalitarian score of her teammates (EgalNi).21

Table 2. Regression analysis: allocated share of credit

(1) (2) (3) (4) (5)

0.1-CoScore 0.3-CoScore 0.5-CoScore 0.7-CoScore 0.9-CoScore

# Teammates /Season 0.00370∗∗∗ 0.00175∗∗∗ 0.000660∗∗∗ 0.0000243 -0.000362∗∗∗

(10.77) (11.02) (7.87) (0.41) (-6.44)

Egali / EgalNi0.160∗∗∗ 0.126∗∗∗ 0.106∗∗∗ 0.0932∗∗∗ 0.0844∗∗∗

(59.22) (101.23) (160.64) (201.75) (190.76)

Constant -0.0873∗∗∗ -0.0449∗∗∗ -0.0199∗∗∗ -0.00433∗∗∗ 0.00597∗∗∗

(-17.00) (-18.93) (-15.87) (-4.94) (7.10)

Observations 3800 3800 3800 3800 3800

Adjusted R2 0.529 0.779 0.904 0.940 0.936

t statistics in parentheses

∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

The regression results indicate that lower values of α generally favor players who have

proved their individual quality by winning (i) with a large number of teammates, and (ii) with

relatively weaker teammates. This can be illustrated by looking at the successful campaign

of the Boston Celtics in the 2007/2008 season. We focus on the ranking of the generalized

21Since there are no non-collaborative projects in this application, the allocated share of credit is equalto Aisi/

∑p∈Ci

w(p).23

α-CoScore for α = 0.3. The five most active players on the roster that season were Paul

Pierce, Ray Allen, Kevin Garnett (dubbed “the big three”), Rajon Rondo and Kendrick

Perkins. Paul Pierce, Ray Allen, and Kevin Garnett improve their ranking significantly,

going from 41th to 25th, 70th to 45th and 34th to 14th, respectively. In contrast, Rajon Rondo

and Kendrick Perkins drop significantly, from 36th to 206th and 608th to 1042th, respectively.

The loss is not surprising for Kendrick Perkins, who has roughly half of the other players’

generalized egalitarian score, and mechanically loses some credit to his stronger teammates.

The loss is more puzzling for Rajon Rondo, who has the second highest generalized egalitarian

score on the team, almost as high as Kevin Garnett’s. However, while Rondo has only played

with the big three (up until 2011), the big three have all played and won games with other

sometimes much weaker teammates. Rondo’s added value to the big three’s success is not

enough to compensate his lack of success outside the team. As a result, Rondo loses some

credit to the benefit of Pierce, Allen, and Garnett.

4. Conclusion

We proposed a new method to quantify individual productivity in the presence of team-

work, CoScore. Our approach is well suited to large scale databases, exploiting variations

in team composition and output. It provides a systematic and objective way of measuring

individual productivity based only on the success of an individual’s teams, information that

can be observed and quantified precisely.

As we have illustrated for academic research and sports, CoScore and its extensions are

practical and can easily be implemented systematically in large databases. Furthermore,

they can be tailored to account for additional information such as the age and the degree of

participation of individuals in their teams. Finally, the inferred credit can be used as an input

for other well known indices that currently do not account for teamwork. For example, in

the case of academic authorship, we could compute the h, i-10, Euclidean (Perry and Reny,

2016) or step-wise (Chambers and Miller, 2014) indices using the credits inferred by CoScore.

In practice, applying CoScore and its extensions to a particular database requires two

important considerations: choosing a measure of team output and selecting an appropriate

extension of CoScore.

Firstly, our analysis takes as input a well defined numerical measure of team output, such

as the profit generated by a management team, the number of citations of an academic

paper, etc. Depending on the application, different numerical measures of team success can

be used, affecting the resulting individual productivity scores and rankings. For example, if

a project is a film, its team is its cast, and its success is measured by box office revenue, the

actors that systematically star in blockbusters will be ranked highly; in contrast, if success24

is measured using an index of critic ratings, artistically accomplished actors will be ranked

highly. An application of CoScore thus first requires specifying what individual productivity

we seek to measure and then selecting the appropriate measure of team success. Our focus in

this paper is on deriving individual productivity given an appropriate team output measure.

A significant amount of research already exists on deriving cardinal values for heterogeneous

objects such as research projects, the value of a win in sports (Hochbaum and Levin, 2006)

or scientific papers (Palacios-Huerta and Volij, 2004; Palacios-Huerta and Volij, 2014); these

measures can all be used as inputs for CoScore and its extensions.

Secondly, depending on characteristics of teamwork in an application, different extensions

of CoScore may be necessary. For example, the generalized α-CoScore productivities and

credits depend on the parameter α specifying the fraction of credit that is allocated ex-

ogenously across team members. As illustrated in our two numerical applications, different

values for α may lead to significant differences in both ranking and scores. It is therefore

important to understand how “best” to choose such a parameter. In the context of academic

coauthorship, a possible strategy consists in estimating the parameter α that best predicts

professional success, as measured by tenure decisions or salaries. Ellison (2013) follows a sim-

ilar approach, investigating which academic productivity index in a family of h- like indices

best matches the professional placement of academic economists. The choice of the param-

eter α may also reflect other considerations, both normative and positive. For example, it

may be desirable to guarantee a minimum level of credit for each individual contributing to

a given project, or to provide individual incentives for team formation.

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Appendix A. Well defined scores

We only need to show that the score in (4) is well defined since this implies the other

results, by considering the case where α = 0, each individual i has age Ai = 1 and has a

binary time participation in each project p, with ti(p) = 1 if i ∈ T (p) and ti(p) = 0 otherwise.28

Theorem 3. The system of equations (4) has a unique solution if either (i) α > 0 or

(ii) each individual has a strictly positive individual contribution.

Proof. If α = 1, then equation system (4) reduces to

si =1

Ai

∑p∈Ci

ti(p)∑j∈T (p)

tj(p)w(p) ∀i ∈ N.

This system trivially has a unique solution. Thus, from here on, consider the case where

α < 1. Let φ : RN++ → RN

++ denote the function defined by setting, for each x ∈ RN++,

(6) φi(x) =1

Ai

∑p∈Ci

α ti(p)∑j∈T (p)

tj(p)+ (1− α)

ti(p)xi∑j∈T (p)

tj(p)xj

w(p) ∀i ∈ N.

We now show that φ has a unique fixed point. Let W =∑

p∈C w(p) and, for each i ∈ N , let

Ii denote the collection of i’s individual projects, all p ∈ C such that T (p) = {i}. For each

i ∈ N , let

Vi =∑

p∈Ci\Ii

ti(p)∑j∈T (p)

tj(p)w(p) and Wi =

∑p∈Ii

w(p).

Under assumption (i), for each i ∈ N , either Wi > 0 or αVi > 0 or both. Under assump-

tion (ii), for each i ∈ N , Wi > 0. Thus, independently of which assumption we make, for each

i ∈ N , αVi +Wi > 0. Note that, for each x ∈ RN++ and each i ∈ N , Aiφi(x) ≥ αVi +Wi > 0

and∑

i∈N Aiφi(x) = W . Thus, letting

K ={x ∈ RN

∣∣∣ ∀i ∈ N, αVi +Wi ≤ Aixi ≤ W},

we can redefine φ as a function mapping K into K. Note that K is compact and convex.

Furthermore, φ is continuous on K. Thus, by Brouwer’s fixed point theorem, φ has at least

one fixed point. We now prove that this fixed point is unique. Let x∗ ∈ K denote any one

of such fixed points. Note that the coordinates of x∗ are strictly positive because K ⊆ RN++.

Thus, by (6), for each i ∈ N , φi(x∗)

x∗i= 1. That is,

(7)∑p∈Ci

w(p)

α ti(p)

x∗i∑j∈T (p)

tj(p)+ (1− α)

ti(p)∑j∈T (p)

tj(p)x∗j

= Ai ∀i ∈ N.

Let f : RN++ → R denote the function defined by setting, for each x ∈ RN

++,29

f(x) ≡ −∑j∈N

Ajxj +∑p∈C

w(p)

α ∑j∈T (p)

ln(xj) ·tj(p)∑

k∈T (p)

tk(p)+ (1− α) ln

∑j∈T (p)

tj(p)xj

.

We first show that f is strictly concave. Let g : RN++ → R be defined, for each x ∈ RN

++,

by g(x) =∑

j∈N [αVj + Wj] · ln(xj). The function g is strictly concave since its Hessian is

negative definite because, for each i ∈ N , αVi + Wi > 0. Let Co ⊆ C consist of all of the

collaborative projects in C. Let h : RN++ → R be defined by setting, for each x ∈ RN

++,

h(x) = −∑j∈N

Ajxj + (1− α)∑p∈Co

w(p) ln

∑j∈T (p)

tj(p)xj

.

As the sum and composition of concave and affine functions, h is concave. Since the sum of

a strictly concave function and a concave function is strictly concave, f = g + h is strictly

concave.

By the strict concavity of f and the fact that RN++ is open and convex, x∗∗ is the unique

maximizer of f over RN++ if and only if ∇f(x∗∗) = 0. That is,

0 =∂f

∂xi(x∗∗) = −Ai +

∑p∈Ci

w(p)

α ti(p)

x∗∗i∑j∈T (p)

tj(p)+ (1− α)

ti(p)∑j∈T (p)

tj(p)x∗∗j

∀i ∈ N

Thus, by (7), x∗ = x∗∗. Recall that x∗ is an arbitrary fixed point of φ. By uniqueness of x∗∗,

we conclude that the fixed point of φ is indeed unique. v

Appendix B. Proof of Theorem 1

Proof of Theorem 1. Let r denote a rule satisfying anonymity and own record only. Let r

denote the egalitarian rule. Since it is obvious that r satisfies the above axioms, it suffices

to prove that r = r. For any rule and any database, only the individual involved in a non-

collaborative project receives credit for it. Thus, the allocation of credit for non-collaborative

projects coincides across rules. Thus, it suffices to show that the allocation of credit of r and

r coincides across collaborative projects.

Step 1: Let D denote a database such that i, j ∈ N(D) and

Ci(D) \ C({i},D) = Cj(D) \ C({j},D).30

Let D ′ denote a database such that N(D) = N(D ′) and

C(D) \ [C({i},D) ∪ C({j},D)] and C(D ′) \ [C({i},D ′) ∪ C({j},D ′)] coincide.

Then, for each p ∈ Ci(D) \ C({i},D), r(p,D) = r(p,D ′).

Let D denote a database where there are i, j ∈ N(D) sharing the same set of collaborative

projects, Ci(D) \ C({i},D) = Cj(D) \ C({j},D) and let p ∈ Ci(D) \ C({i},D). Let D′

be any database such that C(D ′) \ C({j},D ′) and C(D) \ C({j},D) coincide. Since, for

each k ∈ N(D) \ {j}, Ck(D) and Ck(D ′) coincide, by own record only, rk(p,D) = rk(p,D′).

Thus, since the sum of the credits assigned by r to the individuals involved in p adds up to

w(p,D) = w(p,D ′), rj(p,D) = rj(p,D′).

Let D′′ be any database such that C(D ′′) \ C({i},D ′′) and C(D ′) \ C({i},D ′) coincide.

Since, for each k ∈ N(D)\{i}, Ck(D ′) and Ck(D ′′) coincide, by own record only, rk(p,D′) =

rk(p,D′′). Thus, since the sum of the credits assigned by r to the individuals involved in p

adds up to w(p,D ′) = w(p,D ′′), ri(p,D′) = ri(p,D

′′).

Altogether, r(p,D) = r(p,D ′′). Since

C(D) \ [C({i},D) ∪ C({j},D)] and C(D ′′) \ [C({i},D ′′) ∪ C({j},D ′′)] coincide,

this concludes the proof of the Step 1.

Step 2: For each database D, each pair i, j ∈ N(D) such that

Ci(D) \ C({i},D) = Cj(D) \ C({j},D),

and each p ∈ Ci(D) \ C({i},D), ri(p,D) = rj(p,D).

Let D denote a database where there are i, j ∈ N(D) sharing the same set of collaborative

projects, Ci(D)\C({i},D) = Cj(D)\C({j},D). Let p ∈ Ci(D)\C({i},D) and x = r(p,D).

Let D ′ denote a database such that

i. C(D ′) \ [C({i},D ′) ∪ C({j},D ′)] and C(D) \ [C({i},D) ∪ C({j},D)] coincide,

ii. C({i},D ′) and C({j},D) coincide,

iii. C({j},D ′) and C({i},D) coincide.

Let y = r(p,D ′). By Step 1, y = x. By anonymity, yi = xj and yj = xi. Thus, xi = xj.

Thus, ri(p,D) = rj(p,D).

Step 3: For each database D, each i ∈ N(D), and each p ∈ Ci(D), ri(p,D) = w(p,D)#T (p,D)

.

The proof is by induction.

Induction basis: For each D and each i ∈ N(D),

if #Ci(D) \ C({i},D) = 1, then, for each p ∈ Ci(D), ri(p,D) = w(p,D)#T (p,D)

.31

Let D denote a database and i ∈ N(D) be such that Ci(D) contains a single collaborative

project denoted by p. Let D ′ denote a database such that N(D ′) = N(D) and such that

C(D ′) and {p} ∪⋃

j∈N(D)

C({j},D) coincide.

Thus, the only collaborative project in D ′ is p and each individual has the same non-

collaborative projects in D as in D′. By Step 2, for each pair j, l ∈ T (p,D ′), rj(p,D′) =

rl(p,D′). Thus, ri(p,D

′) = w(p,D ′)#T (p,D ′)

. Since Ci(D ′) and Ci(D) coincide, by own record only,

ri(p,D) = ri(p,D′) =

w(p,D ′)

#T (p,D ′)=

w(p,D)

#T (p,D).

Induction hypothesis: Let n denote a natural number. For each D and each i ∈ N(D),

if #Ci(D) \ C({i},D) < n, then, for each p ∈ Ci(D), ri(p,D) = w(p,D)#T (p,D)

.

Let D denote a database and i ∈ N(D) be such that #Ci(D) \C({i},D) = n. We will show

that, for each p ∈ Ci(D) \ C({i},D), ri(p,D) = w(p,D)#T (p,D)

.

Let D ′ denote a database such that N(D ′) = N(D) and such that

C(D ′) and Ci(D) \ C({i},D) ∪⋃

j∈N(D)

C({j},D) coincide.

Thus, each collaborative project in D ′ involves i and each individual has the same non-

collaborative projects in D as in D′. Moreover, Ci(D ′) and Ci(D) coincide.

For each j ∈ N(D), since Cj(D ′) \ C({j},D ′) ⊆ Ci(D ′) \ C({i},D ′), we have #Cj(D ′) \C({j},D ′) ≤ n. Thus, for each j ∈ N(D), there are two possible cases:

Case 1: #Cj(D ′) \ C({j},D ′) < n. By the induction hypothesis,

for each p ∈ Cj(D ′) \ C({j},D ′), rj(p,D ′) =w(p,D ′)

#T (p,D ′).

Case 2: #Cj(D ′) \ C({j},D ′) = n. Then, Cj(D ′) \ C({j},D ′) = Ci(D ′) \ C({i},D ′). Thus,

by Step 2,

for each p ∈ Cj(D ′) \ C({j},D ′), rj(p,D ′) = ri(p,D′).

Let p ∈ Ci(D ′) \ C({i},D ′) and t = #T (p,D ′). Let t1 denote the number of j ∈ T (p,D ′)

such that #Cj(D ′) \ C({j},D ′) < n and t2 denote the number of j ∈ T (p,D ′) such that

#Cj(D ′)\C({j},D ′) = n. Since, each collaborative project in D ′ involves i and i has exactly

n collaborative projects, t = t1 + t2. By Case 1, the rule awards t1 individuals credit w(p,D ′)t

.

By Case 2, the rule awards t2 individuals credit ri(p,D′). Since the total credit awarded

adds up to the project’s worth,

t1 ·w(p,D ′)

t+ t2 · ri(p,D ′) = w(p,D ′).

32

Thus, ri(p,D′) = w(p,D ′)

t. Since Ci(D ′) and Ci(D) coincide, by own record only,

ri(p,D) = ri(p,D′) =

w(p,D ′)

t=

w(p,D)

#T (p,D).

By induction, this establishes Step 3 and concludes the proof of the Theorem. v

Appendix C. Proof of Theorem 2

We introduce two additional axioms implied by the axioms in Theorem 2. These axioms,

stated independently, are useful in the arguments that follow. The first axiom requires

credit on collaborative projects to be allocated proportionally to each individual’s contribu-

tion when the database involves two individuals, each with one individual project and one

collaborative project.

Proportionality: For each D with N(D) = {i, j} and C(D) = {pi, pj, p} such that

T (pi,D) = {i}, T (pj,D) = {j}, and T (p,D) = {i, j}, ri(p,D)

rj(p,D)=w(pi,D)

w(pj,D).

The second axiom reflects the requirement that the name or indexing of projects is irrel-

evant to the credit assignment. Only the project worth and team-membership relations are

taken into consideration.

Neutrality: For each pair D ,D ′, if N(D) = N(D ′) and σ : C(D) → C(D ′) is a bijection

such that, for each p ∈ C(D), w(p,D) = w(σ(p),D ′) and T (p,D) = T (σ(p),D ′), then, for

each p ∈ C(D) and each i ∈ N(D), ri(σ(p),D ′) = ri(p,D).

Remark 2. Aggregate worth implies neutrality. Aggregate strength implies anonymity.

Lemma 1. The CoScore rule satisfies consistency, aggregate worth, aggregate strength, and

proportionality.

Proof. Let r denote the CoScore rule and take a database D .

Consistency

Let k ∈ N(D), x ≡ r(D), and y ≡ r(D−kx ). We need to prove that, for each i ∈ N(D−kx )

and each p ∈ C(D−kx ), xi(p) = yi(p). Using the definitions of x and y in Remark 1, for each33

p ∈ C(D−kx ) ∩ Ck(D) and each i ∈ T (p,D−kx ),

yi(p) = w(p,D−kx )

∑q∈Ci(D

−kx )

yi(q)∑j∈T (p,D−k

x )

∑q∈Cj(D−k

x )

yj(q)

= [w(p,D)− xk(p)]

∑q∈Ci(D)

yi(q)∑j∈T (p,D)\{k}

∑q∈Cj(D)

yj(q)

= w(p,D)

1−

∑q∈Ck(D)

xk(q)∑j∈T (p,D)

∑q∈Cj(D)

xj(q)

∑

q∈Ci(D)

yi(q)∑j∈T (p,D)\{k}

∑q∈Cj(D)

yj(q)

=

w(p,D)

∑j∈T (p,D)\{k}

∑q∈Cj(D)

xj(q)∑j∈T (p,D)

∑q∈Cj(D)

xj(q)

∑

q∈Ci(D)

yi(q)∑j∈T (p,D)\{k}

∑q∈Cj(D)

yj(q)

(8)

and, for each p ∈ C(D−kx ) \ Ck(D) and each i ∈ T (p,D−kx ),

(9) yi(p) = w(p,D−kx )

∑q∈Ci(D

−kx )

yi(q)∑j∈T (p,D−k

x )

∑q∈Cj(D−k

x )

yj(q)= w(p,D)

∑q∈Ci(D)

yi(q)∑j∈T (p,D)

∑q∈Cj(D)

yj(q).

Using the definition of x in Remark 1, for each p ∈ C(D−kx )∩Ck(D) and each i ∈ T (p,D−kx ),

xi(p) = w(p,D)

∑

q∈Ci(D)

xi(q)∑j∈T (p,D)

∑q∈Cj(D)

xj(q)

=

w(p,D)

∑j∈T (p,D)\{k}

∑q∈Cj(D)

xj(q)∑j∈T (p,D)

∑q∈Cj(D)

xj(q)

∑

q∈Ci(D)

xi(q)∑j∈T (p,D)\{k}

∑q∈Cj(D)

xj(q)

(10)

and, for each p ∈ C(D−kx ) \ Ck(D) and each i ∈ T (p,D−kx ),

(11) xi(p) = w(p,D)

∑q∈Ci(D)

xi(q)∑j∈T (p,D)

∑q∈Cj(D)

xj(q).

Let z ∈ Z(D−kx ) be such that, for each p ∈ C(D−kx ) and each i ∈ N(D−kx ), zi(p) = xi(p). Note

that, by (10) and (11), z satisfies the system of equations in (8) and (9). By Remark 1, (8)34

and (9) uniquely define y. Thus, z = y. Thus, for each i ∈ N(D)\{k} and each p ∈ C(D−kx ),

ri(p,D−kx ) = ri(p,D). Thus, r satisfies consistency.

Aggregate worth

Let C ⊆ C(D) be such that for each pair p, q ∈ C, T (p,D) = T (q,D) as in the hypothesis

of aggregate worth. Let S ⊆ N(D) be such that C ⊆ C(S,D). Let s = s(D) and s′ =

s(DC) denote the profiles of scores defined in equation system (3) for databases D and DC

respectively. Replacing all projects C by a single project pC with total equal aggregate worth

in equation system (3), we see that s = s′. Thus, by the definition of r as the CoScore rule,

for each i ∈ S,∑p∈C

ri(p,D) =∑p∈C

w(p,D)si∑

j∈T (p,D)

sj= w(pC ,DC)

s′i∑j∈T (p,DC)

s′j= ri(pC ,DC)

while for each p ∈ C(D) \ C and each i ∈ T (p,D),

ri(p,D) = w(p,DC)si∑

j∈T (p,D)

sj= w(p,DC)

s′i∑j∈T (p,DC)

s′j= ri(p,DC).

Thus, r satisfies aggregate worth.

Aggregate strength

Let S ⊆ N(D) be such that each individual in S has the same list of collaborative projects:

for each pair i, j ∈ S, Ci(D) \ C({i},D) = Cj(D) \ C({j},D). By definition of the CoScore

rule in Remark 1, for each p ∈ C(D) and each i ∈ T (p,D),

ri(p,D) = w(p,D)

∑q∈Ci(D)

ri(q,D)∑j∈T (p,D)

∑q∈Cj(D)

rj(q,D).(12)

Let z ∈ Z(DS) be such that, for each p ∈ C(DS),

ziS(p) =∑i∈S

ri(p,D) and, for each j ∈ T (p,D) \ {iS}, zj(p) = rj(p,D).

Recall that C(DS) = C(D) and that, for each p ∈ C(D), w(p,DS) = w(p,D). Thus, by (12),

for each p ∈ C(DS) and each i ∈ T (p,DS),

zi(p) = w(p,DS)

∑q∈Ci(DS)

zi(q)∑j∈T (p,DS)

∑q∈Cj(DS)

zj(q)

35

which means z satisfies the system of equations defining the CoScore rule endogenously

using (2) and Remark 1 for database DS. By Remark 1, r(DS) = z. Thus, r satisfies

aggregate strength.

Proportionality

Suppose that N(D) = {i, j} and that C(D) = {pi, pj, p} are such that T (pi,D) = {i},T (pj,D) = {j}, and T (p,D) = {i, j}. Then, by equation system (2) and Remark 1,

ri(p,D) =w(pi,D) + ri(p,D)

w(pi,D) + ri(p,D) + w(pj,D) + rj(p,D),

rj(p,D) =w(pj,D) + rj(p,D)

w(pi,D) + ri(p,D) + w(pj,D) + rj(p,D).

Dividing the first equation by the second one and rearranging yields the desired conclusion,

ri(p,D)/rj(p,D) = w(pi,D)/w(pj,D). Thus, r satisfies proportionality. v

Before proving the main theorem, we prove the following result:

Theorem 4. The only rule satisfying consistency, aggregate worth, and proportionality is

the CoScore rule.

Proof. Let r denote a rule satisfying consistency, aggregate worth, and proportionality.

Then, by Remark 2, r also satisfies neutrality. Let r denote the CoScore rule. By Lemma 1,

it suffices to prove that r = r.

Step 1: For each database D such that #N(D) = 2, r(D) = r(D).

Let database D be such that N(D) = {i, j} and let K denote sum of the output of all

collaborative projects in D ; that is, K =∑

p∈C({i,j},D) w(p,D). Let D denote the family of

databases such that each of its members involves the same authors as database D , the non-

collaborative projects are the same as in D , and the sum of the output of the collaborative

projects is also K though the breakup of this aggregate worth into different projects may be

different. Thus, D ′ ∈ D if

- N(D ′) = {i, j};- C({i},D ′) = C({i},D) and C({j},D ′) = C({j},D);

- for each p ∈ C({i},D ′) ∪ C({j},D ′), w(p,D ′) = w(p,D);

-∑

p∈C({i,j},D ′)w(p,D ′) = K.

Note that D ∈ D.

Step 1A: For each pair D ′,D ′′ ∈ D, each p′ ∈ C({i, j},D ′) and each p′′ ∈ C({i, j},D ′′),

w(p′,D ′) = w(p′′,D ′′) implies r(p′,D ′) = r(p′′,D ′′).36

Let all of the notation be the same as in the statement of Step 1A. If D ′ contains a single

collaborative project p′, K = w(p′,D ′) = w(p′′,D ′′). Thus, D ′′ also contains a single col-

laborative project p′′ with worth K. By neutrality, r(p′,D ′) = r(p′′,D ′′). We can repeat

this argument if D ′′ contains a single collaborative project. Thus, from here on, suppose

that each database contains at least two collaborative projects; that is, C({i, j},D ′) and

C({i, j},D ′′) both have at least two elements.

Let D ∈ D be such that C({i, j}, D) = {p′, p}, w(p′, D) = w(p′,D ′), and w(p, D) =

K − w(p′,D ′). By aggregate worth, replacing projects C({i, j},D ′) \ {p′} by p, r(p′,D ′) =

r(p′, D). Similarly, let D ∈ D be such that C({i, j}, D) = {p′′, p}, w(p′′, D) = w(p′′,D ′′),

and w(p, D) = K − w(p′′,D ′′). By aggregate worth, replacing projects C({i, j},D ′′) \ {p′′}by p, r(p′′,D ′′) = r(p′′, D). By neutrality, since the databases D and D differ only in the

indexing of projects,

r(p′,D ′) = r(p′, D) = r(p′′, D) = r(p′′,D ′′).

Step 1B: For each pair p, q ∈ C(D),ri(p,D)

ri(q,D)=w(p,D)

w(q,D).

Let f : [0, K]→ R be such that, for each x ∈ [0, K],

f(x) = ri(p,Dx)

where Dx ∈ D and p ∈ C({i, j}, Dx) is such that w(p,Dx) = x. By Step 1A, f is well

defined. By aggregate worth,

f(x+ y) = f(x) + f(y) ∀x, y ∈ [0, K] such that x+ y ≤ K.

Thus, f satisfies the Cauchy functional equation and there is a constant cK such that, for

each x ∈ [0, K], f(x) = cKx (see Theorem 3 on page 48 of Aczel, 2006). Thus, since D ∈ D,

for each pair p, q ∈ C(D),

ri(p,D) = f(w(p,D)) = cKw(p,D) and ri(q,D) = f(w(q,D)) = cKw(q,D).

This establishes Step 1B.

Step 1C: For each p ∈ C({i, j},D), ri(p,D) =wi(D)

wi(D) + wj(D)w(q,D).

Let D∗ denote the database consisting of a collection of three projects, C(D∗) = {pi, pi, pij},where T (pi,D

∗) = {i}, T (pj,D∗) = {j}, and T (pij,D

∗) = {i, j}. The output of each of these

projects is given by

w(pi,D∗) = wi(D), w(pj,D

∗) = wj(D), w(pij,D∗) = K.

Let B ≡ C({i, j},D).37

By aggregate worth,

(13) ri(pij,D∗) =

∑p∈B

ri(p,D).

By proportionality,

(14) ri(pij,D∗) =

w(pi,D∗)

w(pi,D∗) + w(pj,D∗)w(pij,D

∗).

By Step 1B, for each pair p, q ∈ C(D),

ri(p,D) =w(p,D)

w(q,D)ri(q,D).

Thus, by (13),

ri(pij,D∗) =

∑p∈B

w(p,D)

w(q,D)ri(q,D) =

ri(q,D)

w(q,D)

∑p∈B

w(p,D) =ri(q,D)

w(q,D)w(pij,D

∗).

Thus, by (14),

w(pi,D∗)

w(pi,D∗) + w(pj,D∗)=ri(q,D)

w(q,D).

Thus, for each q ∈ B,

ri(q,D) =w(pi,D

∗)

w(pi,D∗) + w(pj,D∗)w(q,D) =

wi(D)

wi(D) + wj(D)w(q,D) = ri(q,D).

We conclude that r(D) = r(D).

Step 2: For each database D, r(D) = r(D).

Induction hypothesis: Let n denote a strictly positive integer and suppose that, for each D

such that #N(D) ≤ n, r(D) = r(D).

Let D be such that #N(D) = n+1 and x ≡ r(D). For each k ∈ N(D), let Nk ≡ N(D)\{k}.

Let k ∈ N(D). By consistency,

for each p ∈ C(D) \ C({k},D) and each i ∈ T (p,D) \ {k}, ri(p,D−kx ) = xi(p).

Thus, by the induction hypothesis, since #Nk = n,

(15) for each p ∈ C(D) \ C({k},D) and each i ∈ T (p,D) \ {k}, xi(p) = ri(p,D−kx ).

38

Thus, for each p ∈ C(D) \ C({k},D) and each i ∈ T (p,D) \ {k},

xi(p) = ri(p,D−kx ) = w(p,D−kx )

∑q∈Ci(D

−kx )

ri(q,D−kx )

∑j∈T (p,D−k

x )

∑q∈Cj(D−k

x )

rj(q,D−kx )

by definition of r,

= [w(p,D)− xk(p)]

∑q∈Ci(D)

ri(q,D−kx )∑

j∈T (p,D)\{k}

∑q∈Cj(D)

rj(q,D−kx )

by definition of D−kx ,

= [w(p,D)− xk(p)]

∑q∈Ci(D)

xi(q)∑j∈T (p,D)\{k}

∑q∈Cj(D)

xj(q)by (15).

Thus, for each p ∈ C(D) \ C({k},D) and each pair i, j ∈ T (p,D) \ {k},

xj(p)

xi(p)=

∑q∈Cj(D)

xj(q)∑q∈Ci(D)

xi(q).

Repeating the same argument for each k ∈ N(D),

for each p ∈ C(D) and each pair i, j ∈ T (p,D), xj(p) = xi(p)

∑q∈Cj(D)

xj(q)∑q∈Ci(D)

xi(q).

Thus, for each p ∈ C(D) and each i ∈ T (p,D), summing over j ∈ T (p,D),

∑j∈T (p,D)

xj(p) = xi(p)

∑j∈T (p,D)

∑q∈Cj(D)

xj(q)∑q∈Ci(D)

xi(q).

Recall also that x(p) is a credit allocation for project p so∑

j∈T (p,D) xj(p) = w(p,D). Thus,

for each p ∈ C(D) and each i ∈ T (p,D),

xi(p) = w(p,D)

∑q∈Ci(D)

xi(q)∑j∈T (p,D)

∑q∈Cj(D)

xj(q).

Hence, x satisfies the system of equations (2) (using Remark 1) that endogenously defines

the credit allocation recommended the CoScore rule for database D . However, by Remark 1,39

r(D) is the only solution to these conditions endogenously defining the CoScore rule for

database D . Thus, r(D) = x = r(D), as desired. v

Lemma 2. If r satisfies neutrality and aggregate strength, then it satisfies proportionality.

Proof. Suppose that r satisfies neutrality and aggregate strength. By Remark 2, r also

satisfies anonymity. Let D denote the family of databases such that each of its databases has

a collection of projects consisting of exactly one non-collaborative project for each individual

and one collaborative project involving all of the individuals as team-members. Thus, for

each D ∈ D and each i ∈ N(D), there is a project denoted by pi involving only i and a

project denoted by p involving all of N(D).

Let D ∈ D be such that N(D) = {i, j}, V ≡ w(p,D), and W ≡ w(pi,D) + w(pj,D). For

each x ∈ [0,W ], let Dx ∈ D denote a database such that

w(pi,Dx) = x, w(pj,Dx) = W − x, w(p,Dx) = V.

Let a, b, c ∈ [0,W ] be such that a = b+ c. Let D ′ ∈ D be such that N(D ′) = {i′, i′′, j} where

i′, i′′ 6= i, and

w(pi′ ,D′) = b, w(pi′′ ,D

′) = c, w(pj,D′) = W − a, w(p,D ′) = V.

Since a = b+ c, by aggregate strength,

(16) ri(p,Da) = ri′(p,D′) + ri′′(p,D

′).

Let D ′′ ∈ D be such that N(D ′′) = {i, j′, j′′} where j′, j′′ 6= j, and

w(pi,D′′) = b, w(pj′ ,D

′′) = c, w(pj′′ ,D′′) = W − a, w(p,D ′′) = V.

Since c + (W − a) = W − b, by aggregate strength, ri(p,Db) = ri(p,D′′). By anonymity

and neutrality, ri(p,D′′) = ri′(p,D

′). Combining the two previous equations, ri(p,Db) =

ri′(p,D′). Similarly, ri(p,Dc) = ri′′(p,D

′). Thus, by (16),

(17) if a, b, c ∈ [0,W ] and a = b+ c, then ri(p,Da) = ri(p,Db) + ri(p,Dc).

Define f : [0,W ]→ [0, V ] by setting, for each x ∈ [0,W ], f(x) = ri(p,Dx). By (17), for each

triple a, b, c ∈ [0,W ] such that a = b+ c, f(b+ c) = f(b) +f(c). Thus, f satisfies the Cauchy

functional equation and there is a constant γ such that, for each x ∈ [0,W ], f(x) = γx

(Theorem 3, page 48, Aczel, 2006). Since D = Dw(pi,D), ri(p,D) = γw(pi,D). By anonymity

and neutrality, rj(p,D) = rj(p,Dw(pj ,D)) = γw(pj,D). Thus,

ri(p,D)

rj(p,D)=w(pi,D)

w(pj,D).

That is, r satisfies proportionality. v

Lemmas 1 and 2 and Theorem 4 enable a straightforward proof of Theorem 2.40

Proof of Theorem 2. By Lemma 1, the CoScore rule satisfies the properties in Theorem 2.

Conversely, let r denote a rule satisfying these properties. By Remark 2, r satisfies neutrality.

Thus, by Lemma 2, r satisfies proportionality. Thus, by Theorem 4, r is the CoScore rule. v

Appendix D. Logical independence

We now introduce three rules to establish the logical independence of the axioms in The-

orems 1 and 2:

i. The solo rule allocates the worth of each project proportionally to the total worth

of an individual’s solo projects: for each database D , each p ∈ C(D), and each

i ∈ T (p,D),

ri(p,D) =

∑q∈C({i},D)

w(q,D)∑j∈T (p,D)

∑q∈C({j},D)

w(q,D)w(p,D).

ii. The hybrid rule introduced next is a hybrid of the solo and CoScore rules allocating

the worth of a predetermined project using the solo rule while applying the CoScore

rule for every other project. Formally defining the rule requires some notation. Let

q denote a predetermined project and, for each database D , define the following

associated databases:

D−q is such that C(D−q) = C(D)\{q} and, for each p ∈ C(D−q), w(p,D−q) = w(p,D)

and T (p,D−q) = T (p,D). Thus, D−q denotes the database obtained from D by

removing project q.

D+q is such that C(D+q) = {q} ∪ ⋃i∈N(D) C({i},D) and, for each p ∈ C(D+q),

w(p,D+q) = w(p,D) and T (p,D+q) = T (p,D). Thus, D+q denotes the database

obtained from D by removing all collaborative projects except for q if it is

collaborative.

Let r denote the CoScore rule and define the hybrid rule r as follows: for each

database D ,

r(q,D) = r(q,D+q) and r(p,D) = r(p,D−q) for each p ∈ C(D) \ {q}.

Note that if q /∈ C(D), the hybrid rule coincides with the CoScore rule.

iii. We now introduce a rule ensuring that the ratio between the credit of any two indi-

viduals involved in a project is constant. For each individual i that may be involved

in a database, let ci denote a strictly positive scalar and let c denote the vector of such

scalars specifying the corresponding scalar for each individual. We assume that there

are i and j such that ci 6= cj, otherwise the rule will coincide with the egalitarian41

rule. For each such c, the fixed ratio rule (parametrized by c) is defined as follows:

for each database D , each p ∈ C(D), and each i ∈ T (p,D),

ri(p,D) =ci∑

j∈T (p,D)

cjw(p,D).

We emphasize that the parameters specifying a fixed ratio rule are exogenous.

The following table summarizes which axioms are satisfied by each of the rules discussed

thus far.

CoScore Egalitarian Solo Hybrid Fixed ratio

anonymity + + + + −consistency + + − + +

aggregate worth + + + − +

aggregate strength + − + + −own record only − + − − +

Table 3. An axiom in the row is satisfied by a rule in the column (+) or it is not (−).

Proposition 1. Consistency, aggregate worth, and aggregate strength are logically indepen-

dent.

Proof. The egalitarian rule satisfies consistency and aggregate worth but not aggregate

strength. The solo rule satisfies aggregate worth and aggregate strength but not consis-

tency. While the hybrid rule satisfies consistency and aggregate strength, it does not ag-

gregate worth since its predetermined project q cannot be aggregated with other projects

involving the same team without altering the credit allocation for the combined project and

elsewhere in the database. v

Proposition 2. Anonymity and own record only are logically independent.

Proof. The CoScore rule satisfies anonymity but not own record only. The fixed ratio rule

satisfies own record only but not anonymity. This rule is not anonymous since the allocation

of credit is determined by exogenous parameters ci that may be specific to each individual.

It satisfies own record only since the allocation of credit in each project depends only on the

team composition of the project, not on any other information in the database. v42

Appendix E. Data

The productivity of academic researchers. The data was retrieved from Thomson

Reuters’s Web of Science database on February 10th 2016.22 We extracted all published

articles with at least one citation from the following 33 major journals in Economics over

the period 1970− 2015:

Econometrica, Quarterly Journal of Economics, Journal of Economic Literature, American

Economic Review, Journal of Political Economy, Review of Economic Studies, Journal of

Monetary Economics, Journal of Economic Theory, Games and Economic Behavior, Jour-

nal of Economics Perspectives, Journal of Econometrics, Rand Journal of Economics, Eco-

nomic Theory, Journal of Labor Economics, Journal of Human Resources, Journal of Public

Economics, Review of Economics and Statistics, Econometric Theory, Journal of Risk and

Uncertainty, International Economic Review, Journal of Applied Econometrics, European

Economic Review, International Journal of Game Theory, Social Choice and Welfare, Jour-

nal of Environmental Economics and Management, Economic Journal, Journal of Interna-

tional Economics, Journal of Economic Dynamics and Control, Journal of Mathematical

Economics, Economic Inquiry, Scandinavian Journal of Economics, Economics Letters, Ox-

ford Bulletin of Economics and Statistics.

The Web of Science database is not entirely consistent regarding the name of the authors.

Authors are sometimes identified with their first initial only, all their initials and in some

cases their full name. We choose here to retain the first initial only, which prevents the

same author from being identified under two or more labels but means that authors with the

exact same name and first initial are collapsed together. Note that the number of citations

provided by Web of Science is usually smaller than the one of Google Scholar.

For each individual in i the database, let Ii denote her non-collaborative projects, all

p ∈ C such that T (p) = {i} and, for each project p, let ri(p) denote her credit on project p

under the relevant generalized α-CoScore. We can now define the variables used in the OLS

regressions:

- i’s share of credit on her collaborative projects,

∑p∈Ci\Ii

ri(p)∑p∈Ci\Ii

w(p)

22http://ipscience.thomsonreuters.com/product/web-of-science/43

- i’s average number of teammates in her collaborative projects,

1∑p∈Ci

w(p)·∑

p∈Ci\Ii

w(p)

#T (p)

The average is computed by weighting the number of teammates on each paper by

the paper’s worth, then normalizing by the total worth of i’s record.

- i’s share of non-collaborative project worth,∑p∈Ii

w(p)∑p∈Ci

w(p)

- The average egalitarian score of i’s teammates (EgalNi)

EgalNi=

1∑p∈Ci\Ii

#T (p)− 1

#T (p)w(p)

·∑

p∈Ci\Ii

w(p)∑

j∈T (p)\{i}

Egalj

The average is computed by weighting each individual’s egalitarian score by the total

worth jointly produced with i. The first term is a normalizing factor.

44